The risk-return relationship in the South Africa stock market - African ...

The risk-return relationship in the South Africa stockmarketPrepared byLeroi Raputsoane * †South African Reserve Bank(012) 313 3273Leroi.Raputsoane@resbank.co.zaJuly 2009AbstractThis paper analyses the intertemporal risk-return relationship in the South Africanstock market based on Merton’s (1973) single factor ICAPM framework. TheGARCH-in-mean model is used to analyse the daily excess returns of market andindustry stock price indexes of the Johannesburg stock exchange listed companies.The estimated results generally lend support to the robust positive risk-returnrelationship between expected returns and the market risk premium. This suggeststhat the market and industry excess returns in the South African stock market behaveaccording to the standard asset pricing theory.JEL classification: C22, G12Keywords: Intertemporal Capital Asset Pricing Model, risk-return relationshipCorresponding author’s e-mail address: Leroi.Raputsoane@resbank.co.za* The views expressed are those of the author(s) and do not represent those of the South AfricanReserve Bank or the South African Reserve Bank’s policy.† This paper was prepared for the 14th Annual Conference of the African Econometrics Society onEconometric Modelling in Africa, held in July 2009 at the Sheraton hotel, in Abuja, Nigeria.

Table of Contents1 Introduction ..............................................................................................................12 Literature review.......................................................................................................23 Methodology and data .............................................................................................43.1 Model............................................................................................................43.2 Data..............................................................................................................64 Empirical Results .....................................................................................................75 Conclusion................................................................................................................9References ............................................................................................................................11Appendix ...............................................................................................................................13List of TablesTable 1 Variables’ descriptive statistics ..............................................................................7Table 2 GARCH(1,1)-M model results................................................................................9Table A2 Variables’ descriptions.........................................................................................13ii

1 IntroductionStandard asset pricing theory postulates a direct relationship between expectedexcess stock returns and risk. This risk-return trade-off is a long standingphenomenon in investments analysis and is the foundation of financial economics(Leon, Nave and Rubio, 2005). Authors such as Ghysels, Santa-Clara and Valkanov(2004) describe this relation as the “first fundamental law of finance.” The risk-returntrade-off is explained by the Capital Asset Pricing Model (CAPM), which relates therequired return on investment to the risk of undertaking such an investment.Specifically, Merton’s (1973) Intertemporal Capital Asset Pricing Model (ICAPM)hypothesises a positive correlation between expected return on an investment andthe associated risk.The rate of return on an investment is weighted by the perceived risk of undertakingsuch an investment. This implies a direct relationship between market risk and returnfor the reason that risk-averse investors require additional compensation forassuming extra risk. Markets which are perceived by investors to be high risk areassociated with higher returns in order to compensate the risk involved in investing insuch markets. Conversely, lower risk markets are characterised by relatively lowerreturns. Thus it is unambiguous that the risk-return relationship is a fundamentalconcept in investment decision making and that it is accepted as the cornerstone ofrational expectations asset pricing models.This paper examines the intertemporal risk-return relationship in the South Africanstock market based on Merton’s (1973) single factor ICAPM framework. Although it isa long standing phenomenon in investments analysis, the empirical evidence on therisk-return trade-off is ambiguous with some empirical studies documenting a weakor negative relationship at best. Thus there is an ongoing uncertainty in the literatureconcerning this phenomenon and the extent to which the expected return oninvestment is commensurate with the associated risk premium.The next section is the literature review which discusses the theoretical basis and theempirical findings on the risk-return relationship. The discussion on the empirical1

model and data is presented in section 3. Section 4 discusses the empirical results,while section 5 concludes.2 Literature reviewAssuming rational, risk-averse investors, the Sharpe (1964) and Lintner (1965)Capital Asset Pricing Model (CAPM) implies a positive, linear relationship betweenthe expected market risk and returns. Merton (1973) estimated a variant of thetraditional CAPM called the Intertemporal Capital Asset Pricing Model (ICAPM).Similar to the CAPM, the ICAPM implies a positive, linear relationship between theaggregate market return and the market risk premium of the market. This assetpricing model can be summarised by the following equation:E JW ww2wftrm, t1 rf. t1 Etm,t1 Etmf , t1JwJw J [1] where E r r is the conditional market return between period t and t 1.t m, t1f , t1is the rate of return on the stock market index andE r t m, t1r f , t1is the rate of returnon risk-free asset.E2t m , t 1is the conditional variance of the market return or the riskcomponent.E is the covariance of the market return with investmenttmf , t1opportunities or the hedge component. W ( t 1)is wealth and F(t 1) is the vector ofstate variables that describe wealth such thatJ ( W ( t 1),F(t 1),t 1) is the indirectutility function of wealth. JJwwwWis positive if investors are risk averse.is the measure of risk aversion or price of risk andAssuming that investors have a recursive (power) utility function and the rates ofreturn are independent and identically distributed, Merton (1980) agues that relativerisk aversion is constant such that that the hedge component is negligible. Inparticular, Guo and Neely (2006) argue that investment opportunities are sluggish,such that their impact on stock returns is almost constant in the short term. Thismakes the case for the exclusion of the hedge component and the use of high2

frequency data to precisely uncover the risk-return relationship. Therefore, it followsthat: JwwW2Etrm , t1 rf. t1 Et m,t1[2] Jw so that the ICAPM is a single factor model where the conditional excess marketreturn is directly related to the conditional variance of the market. Equation [2] can bewritten as:E r r 2t m, t1f . t10 m m,t1[3]where, as discussed above, mis positive given investor risk aversion and 0isequal to zero.Since Merton’s (1973) work, the single factor risk-return relationship has beensubject to voluminous empirical investigation, with empirical studies foundingconflicting results on the sign and statistical significance of the coefficient of riskaversion (Guo and Whitelaw, 2001). The relationship between market risk-return isfound to be positive and statistically significant by Bollerslev, Engle and Wooldridge(1988), Chou (1988) as well as Harvey (1989). Campbell (1987), Turner, Startz andNelson (1989) as well as Glosten, Jaganathan and Runkle (1993) document anegative and statistically significant relationship, while French, Schwert andStambaugh (1987) as well as Baillie and DeGenmaro (1990) find no statisticallysignificant risk-return trade-off. Nonetheless, a positive, statistically significant riskreturnrelationship has been reinstated by Scruggs (1998) and more recent studiessuch as Guo and Whitelaw (2001), Ghysels, Santa-Clara and Valkanov (2004) aswell as Leon, Nave and Rubio, (2005).It is apparent form the foregoing discussion that the empirical applications of theMerton’s single factor risk-return have achieved mixed successes. The majority of thestudies use several variants of the General Autoregressive ConditionalHeteroscedasticity (GARCH) model to estimate the risk-return relationship. There are3

also instances where Generalised Method of Moments (GMM) is used such as inGuo and Whitelaw (2001). Ghysels, Santa-Clara and Valkanov (2004) introduce theMixed Data Sampling (MIDAS) approach, which forecasts monthly variance with pastdaily squared returns. Although most studies use monthly data, the use of dailydenominated data is also prevalent. Intraday and quarterly data features in somestudies such as Balios (2008) and Guo (2002), respectively. Data span variesconsiderably with Ghysels, Santa-Clara and Valkanov (2004) as well as Lundblad(2004) employing the longest, which cover 76 and 68 years, respectively.3 Methodology and data3.1 ModelFollowing Merton’s (1973) single factor risk-return framework, the following estimableversion of equation [3] is estimated to establish the relationship between market riskpremium and stock returns:2rm, trf , t 0 mm,t m,t [4]where r is the market return and r is the risk-free rate of return so that r r ismthe excess market return.2mfis the variance of excess market return. mis the whitenoise error term, 0is the intercept term and mapproximates the coefficient ofmarket risk aversion. As implied by the theoretical model, the intercept is restricted tozero and the coefficient of market risk aversion is positive. According to Merton(1980), the single factor risk-return model can reasonably approximate the trade-offbetween time-varying risk premium and stock returns and has become the standardmodel in empirical finance literature.mfEquation [4] is estimated directly using the General Autoregressive ConditionalHeteroscedasticity in mean (GARCH-M) model by Engle, Lilien and Robins (1987).This model has received considerable popularity in the risk-return trade-off empiricalliterature because the GARCH type conditional variance is handy as a representation4

of the time-varying risk premium in excess returns. It is also desirable in that itspecifies the heteroskedastic conditional variance term directly into the meanequation so that it characterises the evolution of the excess returns and conditionalvariance simultaneously.The GARCH(p, q )-M model is specified as follows: [4]2rm, trf, t0mm,tm,tqp222m, t jm,tj im,t1j1i1 [5]2m, tm,t1~ N(0,m,t)[6]where equation [4] and [5] are the mean and variance equations, respectively. isthe constant,iand jare the coefficients of the ARCH and GARCH terms,respectively. The market conditional variancevariance based on past informationm.2 mis a one step ahead forecastThe assumption about the distribution of the error term is required because GARCHmodels are computed using the method of maximum likelihood. Gaussian (normal),student’s t and the general error distributions are the three commonly employedassumptions about the distribution of the error term. In this study, the residuals areassumed to be normally distributed as shown in equation [6] above. To ensure thatthe normality condition holds, the Bollerslev and Wooldridge (1992) quasi-maximumlikelihood standard errors are used. The vector of population parameters to beestimated is:2 ( ,,, , , )[7]In estimating the population parameters , the following conditional log likelihoodfunction is maximised under the assumption of normally distributed errors:5

TT21 2 L( ) L t( ) log( ) log(2) [8]2t1 t12 where the first two conditional moments should be correctly specified to ensure thatthe parameters estimation is consistent.3.2 DataAnderson and Bollerslev (1997) argue that the use of high frequency data isdesirable to uncover the risk-return relationship. This is because it allows for a bettermeasurement of risk and enables precise identification of the risk-return trade-off.This is because high frequency data produces better estimates of conditionalvolatility process. The risk-return trade-off is estimated using daily returns on 50market and industry stock price indexes of the Johannesburg stock exchange listedcompanies. These market and industry stock price indexes are weighted by marketcapitalisation, where industry stock price indexes assume the Industry ClassificationBenchmark (ICB) system.The bond exchange yields on R153 (short term government bond) and R186 (longterm government bond) are used to approximate the risk-free rate of interest. Moststudies use the three month treasury bill rate for this purpose. However, the risk freerate of return is only available in monthly frequency. All data are sourced from theSouth African Reserve Bank database and spans the period January 04, 1995 toFebruary 06, 2009. This yields 2635 data points.The data for the industry stock price indexes of aerospace and defence, personalcare and household products, tobacco, utilities, electricity, gas distribution, gas,water, and multiutilities as well as alternative exchange were not available and assuch, their risk-return relationships could not be estimated. The market and industrialgroups’ stock indexes descriptions are detailed in table A1 in the appendix and theirdescriptive statistics are shown on in table 1 below. According to the descriptivestatistics, consumer goods, food producers, equity investments as well asdevelopment and venture capital stock price indexes show high volatility during thesample period based on standard deviations.6

Table 1Variables’ descriptive statisticsVariable Std. Dev. Skewness Kurtosis Variable Std. Dev. Skewness KurtosisS01 1.578 -0.010 7.813 S30 2.209 1.197 17.897S02 2.095 0.308 8.466 S32 1.507 -0.357 7.156S03 2.168 0.323 8.269 S33 1.661 -0.417 6.970S04 2.758 0.611 7.921 S34 1.665 0.106 8.410S06 3.072 -13.76 444.1 S35 2.300 -0.116 6.368S07 2.543 -0.150 6.062 S36 1.798 -0.104 6.735S08 2.805 0.697 8.188 S37 2.538 -22.68 885.7S09 2.688 -0.231 10.20 S38 2.483 0.407 7.804S10 2.470 0.259 6.219 S39 1.821 -0.237 6.726S11 1.477 -0.168 7.420 S40 2.575 0.406 8.301S12 2.046 0.291 9.355 S45 1.636 0.019 7.240S13 1.403 0.148 6.314 S46 2.071 0.201 6.141S14 2.608 -21.81 843.7 S47 1.695 0.032 7.437S15 2.493 0.231 6.584 S48 1.907 -0.083 7.149S16 2.758 0.199 9.808 S49 17.81 49.67 2525.2S17 1.567 -0.248 7.343 S50 1.288 -0.130 7.892S19 1.707 -0.028 7.182 S51 1.872 -0.050 6.784S20 1.555 -0.188 6.933 S52 2.348 -0.280 9.304S21 1.684 -0.695 48.82 S53 1.836 0.082 5.335S22 17.84 49.54 2517.1 S54 2.439 -0.233 8.807S23 2.346 4.193 176.7 S55 22.34 40.08 1686.9S25 1.606 -0.145 6.384 S56 20.06 39.39 1647.9S26 1.935 0.026 6.410 S58 1.284 -0.466 9.781S27 17.70 49.95 2544.8 S59 1.104 -0.662 10.35S28 1.956 0.237 6.154 S62 1.677 0.050 7.5794 Empirical ResultsThe maximum likelihood estimates of the ICAPM are estimated using theGARCH(1,1)-M model for the whole sample period. The data series for some of theindustry stock price indexes do not span the sample period and as such, their riskreturnrelationships were estimated using data from January 04, 1999 up toDecember 30, 2005, April 03, 2007 and December 21, 2007 for mining finance,technology and hardware as well as diamonds and gemstones industries,respectively, while the sample period for development capital and venture capitalspans January 02, 2002 to February 06, 2009.7

In accordance with ICAPM theory, the intercept term was restricted to zero in all theestimated GARCH-M models. Lanne and Saikkonen (2006) have also shown that theinclusion of an intercept term when it is statistically insignificant results in lack ofpower in the standard Wald test on the mean effect parameter in the GARCH-Mmodel. Furthermore, the Bollerslev and Wooldridge (1992) quasi-maximum likelihoodstandard errors and the heteroscedastic consistent covariances are used to ensurethat the normality condition holds in all the estimated models.The maximum likelihood estimates of the GARCH-M model for the coefficient of riskaversion mare reported in table 2. According to the empirical results, out of the 50stock price indexes, 45 (95 percent) show a positive and a highly statisticallysignificant coefficient of risk aversion. This includes the stock price indexes for theoverall market index and the indexes of the major industries such as resources,Industrials, consumer goods, health care, telecommunications and financials. This isalso the case with the major group indexes such as the top 40, mid cap and smallcap stock indexes.The stock price indexes for technology sub-sectors; software and computer servicesas well as technology and hardware are borderline statistically significant. Only 5 (5percent) of the 50 stock price indexes show no statistically significant coefficient ofrisk aversion. These are the indexes for diamonds and germstones, construction andmaterials as well as technology, while the indexes for automobiles and parts as wellas industrial transportation are not only statistically insignificant but also shownegative coefficient of risk aversion.In summary, the empirical evidence on risk-return relationship as conjectured by theICAPM is ambiguous with some empirical studies documenting a weak or negativerelationship at best. In spite of this, the estimated results generally lend support tothe robust positive risk-return relationship between expected returns and the marketrisk premium in the South African stock market, notwithstanding the few exceptions.Consequently, the South African stock market indexes conform to the Merton’s(1973) ICAPM theoretical hypothesis so that the expected return on an investment inthe South African stock market is commensurate with the market risk premium.8

Table 2GARCH(1,1)-M model resultsm Std. Error z-Stat. Variable m Std. Error z-Stat.VariableS01 0.062 0.013 4.905 S30 0.031 0.008 3.732S02 0.042 0.010 4.289 S32 0.054 0.012 4.630S03 0.039 0.009 4.129 S33 0.040 0.011 3.686S04 0.014 0.007 2.061 S34 0.042 0.012 3.623S06** 0.004 0.007 0.628 S35 0.025 0.008 3.079S07 0.026 0.008 3.405 S36 0.036 0.011 3.414S08 0.026 0.007 3.833 S37** -0.002 0.004 -0.404S09 0.016 0.007 2.294 S38 0.030 0.008 3.847S10 0.029 0.008 3.558 S39 0.040 0.010 3.851S11 0.064 0.013 5.077 S40 0.023 0.008 3.070S12 0.040 0.010 4.092 S45 0.042 0.011 3.797S13 0.062 0.013 4.604 S46 0.031 0.009 3.563S14** 0.004 0.003 1.285 S47 0.044 0.011 4.097S15 0.021 0.008 2.724 S48 0.025 0.010 2.489S16 0.021 0.007 3.062 S49 0.006 0.000 20.022S17 0.062 0.013 4.909 S50 0.059 0.015 3.970S19 0.056 0.011 5.189 S51 0.029 0.010 3.004S20 0.044 0.012 3.578 S52** 0.009 0.007 1.210S21 0.055 0.015 3.711 S53 0.021 0.011 1.970S22 0.006 0.000 18.335 S53 0.021 0.011 1.970S23** 0.000 0.010 -0.012 S55 0.005 0.000 16.591S25 0.046 0.012 3.945 S56 0.001 0.000 2.796S26 0.032 0.010 3.100 S58 0.076 0.013 5.770S27 0.006 0.000 93.128 S59 0.100 0.016 6.236S28 0.041 0.009 4.334 S62 0.057 0.012 4.765** Statistical insignificance at 5 percent level5 ConclusionThis study examined the intertemporal risk-return relationship in the South Africanstock market based on Merton’s (1973) single factor ICAPM framework. TheGARCM-M model by Engle, Lilien and Robins (1987) is used to estimate the riskreturntrade-off of 50 daily excess returns of market and industry stock price indexesof the Johannesburg stock exchange listed companies. According to the empiricalresults, 95 percent of stock price indexes show a positive and a highly statisticallysignificant coefficient of risk aversion, while 5 percent are not only statistically9

insignificant but also show negative coefficient of risk aversion. This suggests that,generally, the market and industry stock prices in the South African stock marketconform to the Merton’s (1973) ICAPM theoretical hypothesis of a positiverelationship between excess market returns and the market risk premium.Several challenges to successfully estimating the risk-return trade-off remain andneed further examination. These are the issues of the methodological approach,volatility characteristics of the risk premium as well as data span and frequency. Guoand Neely (2006) suggest that the leverage effects should be accounted for toaddress the asymmetry in the response to the conditional volatility. They furtherargue that the long run conditional variance is more important in the determination ofequity risk premium, which calls for the component GARCH model to uncover theseshort and long run conditional variance dynamics. There is also a need to explorethe MIDAS approach by Ghysels, Santa-Clara and Valkanov (2004), which hasrecently become popular in uncovering the market risk-return trade-off to see if it canreplicate the results in this paper.10

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AppendixTable A2 Variables’ descriptionsShare/Bond Description Share/Bond Descriptionb40 R 153 Government bond s28Healthcare equipment andservicesb42 R 186 Government bond s30 Pharmaceutical and Biotechnologys01 All Share s32 Consumer servicess02 Resources s33 General retailerss03 Mining s34 Travel and leisures04 Gold mining s35 Medias06 Diamonds & gemstones s36 Support servicess07 Platinum & precious metals s37 Industrial transportations08 General mining s38 Telecommunicationss09 Mining finance s39 Food and drug retailerss10 Oil & gas producers s40 Fixed line telecommunicationss11 SA all share industrials s45 Financialss12 Basic materials s46 Bankss13 Chemicals s47 Non-life insurances14 Construction & materials s48 Life insurances15 Forestry and paper s49 Equity investment instrumentss16 Industrial metals s50 Real estates17 Industrials s51 General financials19 General industrials s52 Technologys20 Electronic and electrical equipment s53 Technology and hardwares21 Industrial engineering s54 Software and computer servicess22 Consumer goods s55 Development capitals23 Automobiles and parts s56 Venture capitals25 Health care s58 Mid Caps26 Beverages s59 Small Caps27 Food producers s62 Top 4013